THE  WILEY  TECHNICAL  SERIES 

FOR 

VOCATIONAL  AND  INDUSTRIAL  SCHOOLS 

EDITED  BY 

JOSEPH  M.  JAMESON 

GIRARD  COLLEGE 


PREPAEATOEY  MATHEMATICS 

FOR  USE   IN 

TECHNICAL  SCHOOLS 


THE  WILEY  TECHNICAL  SERIES 

EDITED    BY 

JOSEPH  M.  JAMESON 


MATHEMATICS  TEXTS 

Mathematics  for  Technical  and  Vocational  Schools. 

By  Samuel  Slade,  B.S.,  C.E.,  and  Louis  Margolis, 
A.B.,  C.E.  491  pages.  5i  by  8.     353  figures.      Cloth. 

Mathematics  for  Machinists. 

By    H.    VV.    BuRNHAM,    M.A.     229   pages.     5   by   7. 

175  figures.     Cloth. 

Arithmetic  for  Carpenters  and  Builders. 

By  R.  BuRDETTE  Dale,  M.E.     231  pages.     5  by  7. 
109  figures.     Cloth. 

Practical  Shop  Mechanics  and  Mathematics. 

By   James   F.   Johnson.      130   pages.     5  by   7.     81 
figures.     Cloth. 


CASS  TECHNICAL  HIGH  SCHOOL  SERIES 

Mathematics  for  Shop  and  Drawing  Students. 

By  H.  M.  Keai,  and  C  J.  Leo.nard.  213  pages. 
4Jby7.      188  figures.      Cloth. 

Mathematics  for  Electricel  Students. 

By  II.  M.  Keai,  and  C.  J.  Leonard.  230  pages. 
4  J  by  7.      lti.j  figures.      Cloth. 

Preparatory  Mathematics  for  Use  in  Technical  Schools. 
By  Harold  B.  Ray  and  .Arnold  V.  Doub.  68  pages. 
4  J  by  7.     70  figures.     Cloth. 

Preparatory  Mathematics  for  the  Building  Trades. 

By  Harold  B.  Ray,  .Arnold  V.  Doub,  and  O. 
Frank  Carpenter.  05  pages.  41  by  7.  58  figures. 
Cloth. 

Technical  Mathematics.     Vol.  I. 

By  II.  M.  Keal,  N.  S.  Phelps  and  C.  J.  Leonard. 
231  pages.     45  by  7.      145  figures.     Cloth. 

Technical  Mathematics.     Vol.  II. 

271    pages.     4i    by    7.     306    figures.     Cloth. 

Technical  Mathematics.     Vol.  III. 

138  pages.      4  5  by  7.      136  figures.    Cloth. 

Tables  for  Technical  Mathematics. 

By  H.  M.  Keal,  N.  S.  Phelps  and  C.  J.  Leonard. 
85  pages.     4  J  by  7.     Cloth. 


PREPARATORY    MATHEMATICS 

FOR  USE  IN 

TECHNICAL  SCHOOLS 


BY 
HAROLD  B.  RAY 

AND 

ARNOLD    V.   DOUB 

INSTRUCTORS  IN  MATHEMATICS,  CASS   TECHNICAL  HIGH  SCHOOL 
DETROIT 


NEW  YORK 
JOHN  WILEY  &  SONS,  Inc. 

London:    CHAPMAN  &  H.ALL,    Limited 


Copyright,  1921 

BY 

HAROLD  B.  RAY  and  ARNOLD  V.  DOUB 


All  Rights  Reserved 
This  book  or  any  pari  thereof  must  not 
be    reproduced    in    any   form    without 
the  written  permission  of  the  pxiblisher. 


Printed  in  U.  S.  A. 


9/30 


PRESS    OF 

BRAUNWORTH    &    CO,,    INC. 

BOOK    MANUFACTUREHa 

BROOKLYN,    NEW  YORK 


Kq 


PREFACE 


^  A  LAEGE  number   of    continuation  and  evening  school 

^     students,  who  wish  to  take  up  technical  courses,  are  deficient 
^     in  the  knowledge  of  certain  fundamentals  of  elementary 
3     mathematics.     Most  important  of  these  are  the  rules  and 
^     methods  used  in  solving  problems  involving  common  frac- 
^     tions,  decimals,  and  square  root.     A  short,  yet  thorough 
j^  review  of  these  topics  is  necessary  before  such  students  can 
^   do  satisfactory  work,  either  in  the  shop  and  drafting  room, 
^    or  in  the  accompanying  more  advanced  mathematics, 
t*-  Applications  of  these  basic  operations  in  solving  prob- 

lems of  mensuration,  threads,  gears,  the  micrometer,  and 
^—  percentage,  suggest  and  anticipate  the  problems  of  the 
^  trade  courses. 

Lu  The  material  and  methods  of  this  brief  text  have  been 
"^  selected  after  careful  study,  in  the  class  room,  of  the 
needs  of  several  thousand  students  taking  this  preparatory 
course  in  Cass  Technical  High  School.  Each  discussion  and 
exercise  has  passed  the  test  of  "selling  itself"  to  the  student 
doing  individual  work. 

H.  B.  R. 
A.  V.  D. 
Detroit,  Mich.,  1921. 


^p:if^29 


TABLE   OF  CONTENTS 


CHAPTER  I 

PAGE 

Fractions '- 

Definition.  Reduction  of  Fractions.  Equivalent  Fractions. 
Addition.  Least  Common  Denominator.  Subtraction.  Mul- 
tiplication. Areas.  Division.  Applied  Problems.  Screw 
Threads.     Gears. 


30 


CHAPTER  II 

Decimals 

Definition.  Reading.  Addition.  Subtraction.  Multipli- 
cation. Division.  Decimal  Equivalents.  Percentage. 
Efficiency.  The  Micrometer.  Taper.  Circles.  Cutting 
Speed.     Volume  of  Solids. 

CHAPTER  III 

Powers  and  Roots ^° 

Powers.  Roots.  Square  Root.  Right  Triangles.  Review 
Problems. 

Appendix "•'^ 


Index 


67 


MULTIPLICATION  TABLE 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

1 
12 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

2 

4 

6 

8 

10 

12 

14 

16 

18 

20 

22 

24 

3 

6 

9 

12 

15 

18 

21 

24 

27 

30 

33 

36 

4 

8 

12 

16 

20 

24 

28 

32 

36 

40 

44 

48 

5 

10 

15 

20 

25 

30 

35 

40 

45 

50 

55 

60 

6 

12 

18 

24 

30 

36 

42 

48 

54 

60 

66 

72 

7 

14 

21 

28 

35 

42 

49 

56 

63 

70 

77 

84 

8 

16 

24 

32 

40 

48 

56 

64 

72 

80 

88 

96 

9 

18 

27 

36 

45 

54 

63 

72 

81 

90 

99 

108 

10 

20 

30 

40 

50 

60 

70 

80 

90 

100 

110 

120 

11 

22 

33 

44 

55 

66 

77 

88 

99 

110 

121 

132 

12 

24 

36 

48 

60 

72 

i 

84 

96 

108 

120 

132 

144 

PREPARATORY  MATHEMATICS 


CHAPTER  I 


FRACTIONS 


1.  Definitions. — The  line  from  A  to  B,  or  simply  the 
line  AB,  Fig.  1,  is  four  inches  in  length.  It  is  divided  at 
C  into  two  equal  parts,  AC  and  CB,  each  two  inches  in 
length.     Each  of  these  parts  is  one-half  of  AB,  or  two  inches 


< 



4 

„ 

r 

c 
Fig.  1. 


is  one-half  of  four  inches.  AC  is  divided  at  D  into  two  equal 
parts,  AD  and  DC,  and  each  part  is  one  inch  in  length. 
CB  is  divided  similarly  at  E.  There  are  thus  four  equal 
parts  or  divisions  of  AB,  each  one  inch  in  length,  and  each 
part  is  one  fourth  of  AB. 

That  is,  AZ)  =  i  of  AB;  DC  =  l  of  AB; 

CE^loiAB;  EB  =  loiAB. 


2  FRACTIONS 

Then,     AD-\-DC  =  l  of  AB-\-l  of  AB, 
or,  AC  =  ioiAB. 

Also,  AE  =  i  o(  AB,  and  Ai5  =  t  of  AB. 

Each  of  the  numbers,  |,  j,  f,  f,  and  t,  is  a  fraction. 
The  number  above  the  line,  as  1,  2,  3,  is  the  numerator 
of  the  fraction.  The  number  below  the  line,  4,  is  the 
denominator  of  the  fraction.  The  numerator  and  denomi- 
nator are  the  terms  of  the  fraction. 

2.  Reduction  of  Fractions. — A  fraction  is  reduced  to  its 
lowest  terms  when  the  numerator  and  denominator  taken 
together  can  be  divided  by  no  number  except  1. 

In  Section  1  it  is  stated  that  AC  =  \  of  AB;  also  that 
AC^ioi  AB. 

Thenf  of  A5  =  iof  ^5,  orf  =  i 

If  both  terms  of  the  fraction  f  are  divided  by  2,  the 

resulting  fraction  is  \. 

2-^2_l 
4--2~2' 

A  factor  of  a  number  is  a  divisor  of  the  number. 

One  is  the  only  common  factor  of  1  and  2,  therefore 
the  fraction  f ,  when  reduced  to  ^,  is  expressed  in  its  lowest 
terms. 

Similarly : 

6^32       4^41       15-^53.      24h-2^12,      12-^3^4 
9^'"3'      8T4~2'     25-^5~5'      30^2~15'       15^3~5" 

Sometimes  the  common  divisor  of  the  two  terms  is 
written  before  the  fraction,  thus  3  |  t2  =f . 


REDUCTION  OF  FRACTIONS 


EXERCISE  1 

Reduce  the  following  fractions  to  their  lowest  terms: 

5 
6 


1.  H 

2.  -A 


3        12^ 
•    T8-  T 
4.       20.^ 


T3"-  ? 

4  0     /'^ 

6  4-  :?^ 


7       2.8     /^ 


9       5  6 


8.  ^^A 


3  3"- 


10         48 


11.  The  line  AB,  Fig.  2,  is  divided  into  sixteen  equal 
parts.     How  many  eighths  in  four  of 


these  parts? 


16^ 


^8 


iii. 


I  M  I  I  I  I  I  I 


Fig.  2. 


How  many  eighths  in  twelve  parts? 

How  many  fourths  in  four  parts?     In  twelve  parts? 


12. 


Fig.  3. 


The  circle  of  Fig.  3  is  divided 
into  twelve  equal  parts.  How 
many  fourths  of  the  circle  in 
nine  of  these  parts?  How 
many  thirds  in  eight  parts? 
How  many  sixths  in  ten  parts? 


3.  Equivalent  Fractions. — If  both  terms  of  the  fraction 
\  are  multiplied  by  2,  the  result  is  f . 

1X2^2 
2X2    4* 

Similarly,  any  fraction  can  be  changed  to  an  equivalent 
fraction  by  multiplying  both  numerator  and  denominator 
by  the  same  number. 

2X3_6         4><7_28.        3X8  _24 
3X3~9'       5X7~35'        11X8~88' 


Examples.- 


FRACTIONS 


It  is  often  necessary  to  change  a  fraction  into  an  equiva- 
lent fraction  having  a  specific  number  for  the  denominator. 
This  is  done  as  in  the  following  example : 

Example. — Change  f  to  thirty-seconds;  or  f=3^. 
Divide  32  by  4,  the  result  is  8.  Multiply  3  by  8,  the  result 
is  24,  the  numerator  of  the  new  fraction.     Therefore  f  =tI- 


EXERCISE  2 

1. 

4-32- 

3. 

2  _     ?                        K           4  _     ? 
5  —  40-                   "•         7—  "2  8' 

7. 

5     _     ? 
T2— T8"- 

2. 

1  _     ? 

4. 

7         ■'                 e      1  1  _    ■? 
S  —  6T-               O.    T6—T2- 

8. 

3  _     ? 
8  —  3^ 

I  I  1 1  I  1 1  I  I 


1 ?_. 

4  —  16, 


10. 


Fig.  4. 


The  line  AB,  Fig.  4,  is  di- 
vided into  sixteen  equal  parts. 

How  many  sixteenths  in  j  of 
the  line? 


ITS, 


5  __L-. 
S  —  1  6; 


7  _     ?     .         1  _     ? 

8  — TB"j         S  — Tff- 


The  circle,  Fig.  5,  is  divided 
into  12  equal  parts^  How  many 


twelfths  in   |  of  |fce   circle? 
i?    In  I?     In  f  ? 


In 


Fig.  5. 


4.  Addition  of  Fractions. — In  addition  of  fractions,  the 
fractions  to  be  added  must  first  be  changed  to  fractions 
having  the  sa7ne  number  for  a  denominator. 

This  number  is  called  the  Common  Denominator.  After 
the  common  denominator  is  selected,  the  next  step  is  similar 
to  the  work  done  in  Exercise  2. 


ADDITION  5 

Example  1. — Add  ^  and  ^. 

The  common  denominator  must  be  a  number  divisible 
by  the  two  denominators.  6  and  12  are  two  numbers 
divisible  by  both  3  and  2.  It  is  advisable  to  use  the  smallest 
number  possible;  this  number  is  called  the  Least  Common 
Denominator.     (Abbreviated  L.  C.  D.) 

After  selecting  6  for  the  L.  C.  D.,  change  ^  and  ^  to 
sixths. 

•  1  —  2. 

3  ~  6  > 

1  _  3.. 

2  —  6  , 

f+t=-|-     Ans. 

The  last  step  is  performed  by  adding  the  numerators 
and  writing  the  sum  over  the  common  denominator. 

When  the  denominators  are  small  numbers,  the  L.  C.  D. 
is  easily  obtained  bj^  reference  to  the  multiplication  table. 
If  this  is  not  satisfactory,  the  product  of  the  denominators 
may  be  used  as  the  common  denominator,  although  this 
method  will  not  always  give  the  least  common  denominator. 
Instructions  will  be  given  later  for  finding  the  L.  C.  D. 
when  the  denominators'  are  large  numbers. 

Example  2.— i  +  f +f  =  ? 

In  the  table  of  6's  (see  table  facing  page  1),  72  is 
the  smallest  number  in  the  table  that  is  exactly  divisible 
by  both  8  and  9.  Then  it  is  the  least  common  denom- 
inator for  the  three  fractions.  Arrange  the  solution  as 
follows : 


3  _  27 

S  ~  72 

2  _  JJ8. 
■ff  —  72 


The  sum  of  the  numerators  (12+27+16) 
is  55;  the  sum  of  the  fractions  is  rl. 

Ans. 


FRACTIONS 
Example  3.— 1+^-|-^  =  ? 


2  _  28 

3  =¥2 

1  ^  |i.  The  product  of  the  denominators  (3X2X7) 

iL— 2j4         is  42.     The  sum  of  the  numerators  is  73;   the 

7  —  42  ' 


sum  of  the  fractions  is  if  • 


7|=lfi    Ans. 

A  fraction  whose  numerator  is  larger  than  the  denomi- 
nator is  an  Improper  Fraction.  Such  a  fraction  should  be 
reduced  by  dividing  the  numerator  by  the  denominator. 
734-42=1,  with  a  remainder  of  31.  The  result  is  written, 
Ifi.  This  is  a  mixed  number.  A  Mixed  Number  is  a 
number  made  up  of  a  whole  number  and  a  fraction. 

5.  Addition  of  Mixed  Numbers. — If  mixed  numbers,  or 
mixed  numbers  and  fractions,  are  to  be  added,  add  the 
whole  numbers  and  fractions  separately,  and  combine  the 
results. 

Example.— 81+ t+12f  =  ? 

82  _      O  2.0. 
3  —      0  3  0 

1=     U        The  L.  C.  D.  is  30. 
12f=mf 


^3lf 


6  3  _  9_3_  _  9   1_  . 
3~0  —  •^  3  0  —  ^ITU  • 

20+2Tio=22TV.     Ans. 


EXERCISE  3 


1.  i+|  =  ?         3.  RH?       5.     i+f  =  ?     7.  2f+8f  =  ? 

2.  i  +  |  =  ?         4.  i^h  =  '^       6.  U+24  =  ?   8.  4|+Gf  =  ? 


ADDITION 
9.  Find  the  over-all  dimension  of  Fig.  6. 


,// 

_ 

r  II                         < 

p< 

2\ 

"^ 

'10                   ^ 

1 

i 

is 

I 

7 

1 
^1 

I 

Fig.  6 


10.  Find  the  over-all  dimension  of  Fig.  7. 


Fig.  7. 


11.  Measure  the  length  of  the  lines  indicated  in  Fig.  8, 
add,  and  check  by  measuring  the  over-all  length. 


Note. — In  problems  11  and  12  the  student  should  use  a  machinist's 
scale  or  a  rule  with  yj  inch  divisions. 


8  FRACTIONS 

12.  Measure  the  lines  of  Fig.  9  and  check  as  in  Prob.  11. 


I  ■  ■.      I    ■        r       I 


Fig.  8. 


6.  To  Find  the  Least  Common  Denominator. — In  some 
cases,  the  least  common  denominator  can  not  be  found 
easily  by  the  methods  explained  in  Section  4. 


Fia.  9. 


LEAST  COMMON  DENOMINATOR  9 

Example.— Find  the  L.  C.  D.  of  t^,  -h,  and  i^.     The 

following  method  is  generally  used. 

2)12     21     14 

7)  6     21      7 

3)  6      3       1 


2       1       1         2X7X3X2X1X1  =  84,L.C.D. 

Rule. — Divide  the  denominators  by  any  number  that  will 
divide  exactly  two  or  more  of  them,  bringing  down  the  denomi- 
nators that  cannot  be  exactly  divided  by  the  divisor.  Repeat  the 
operation  until  no  two  numbers  left  are  exactly  divisible  by  the 
same  number.  The  product  of  the  remaining  numbers  multi- 
plied by  the  divisors  will  be  the  Least  Common  Denominator. 

EXERCISE  4 

Add: 

1.  6i  +71  +  32^^.  Ans.  17,^. 

2.  4i  +56^4+  6f.  Ans.  16M. 

3.  7f  +2f  +  9^.  Ans.  19iff. 

4.  -V-+  ¥+  tV.  Ans.  12i. 

5.  n  +4TV+18f.  Ans.  33 iV- 

6.  4f  +7i  +  9i.  Ans.  21f . 

7.  8|  +32^4+  8i^.  Ans.  20^. 

8.  2TV+lf  +  BiV.  Ans.  7^. 

9.  How  many  rods  of  fencing  will  it  take  to  inclose  a 
lot  the  sides  of  which  measure  10|  rods,  17f  rods,  12t^ 
rods,  and  8f  rods?  Ans.  493^^  rods. 

10.  A  machinist  wishes  to  cut  three  lengths  from  a  stock. 
The  first  length  must  be  If",  the  second  2^6",  and  the  third 
96V"-  How  long  must  the  stock  be  to  furnish  the  required 
lengths?  Ans.  12|f". 


10 


FRACTIONS 


11.  Find  the  missing  dimensions  in  Figs.  10  and  11. 


Mi 


Fig.  10. 


if^lfe'^ 


1 

1                ] 

1 

1 

1 

i 

1 

i         „         i 

rf ? ^t* -? >1 

Fig.  11. 

12.  Find  the  total  length  of  a  generator  shaft  if  12f 
inches  are  allowed  for  the  armature,  4t  inches  for  the  bear- 
ings, SiTT  inches  for  the  commutator,  and  1^  inches  are 
clear.  Ans.  211^  inches. 

7.  Subtraction  of  Fractions. — In  the  subtraction  of  frac- 
tions, as  in  addition,  the  given  fractions  must  be  changed 
to  equivalent  fractions  having  a  common  denominator,  then 
the  difference  of  the  numerators  can  be  found. 

Example  1.  Example  2. 

l_i    —   ?  12^—  ^i   —   ? 


The  L.  C.  D.  is  6. 


The  L.  C.  D.  is  24. 


2  _  1 


Oc  —     Oo 


2T 


Ans. 


'^T- 


Ans. 


-k 


SUBTRACTION 

EXERCISE  5 

Subtract  the  following: 
1.     2f                3.  31t^ 

5.    91 

Si 

7.  1631 
97f 

2.  131 

4.          ^ 

*•               3 

3 

8 

6.  3  If 

4i 

8.     24f 

19f 

11 


9.  If  2f  inch  screws  are  used  to  fasten  porcelain  insulators 
on  a  wall,  and  the  insulators  are  lye  inches  thick,  how  far 
will  the  screws  penetrate  the  wall? 

10.  Find  the  missing  dimensions  in  Figs,  12  and  13. 


Fig.  12. 


Fig.  13. 


11.  Find  the  lengths  indicated  in  Fig.   14,  (a),  (6),  (c), 
and  (d)  by  subtraction,  and  check  each  by  measuring. 

Examples.— 5f- If  =  ? 
The  L.  C.  D.  is  8. 

5t  =  5|=4|+|=4i^  Check: 

13  =  lf=.lf        =lf  Does,  lf+3| 

31.     Ans. 


■5f? 


12 


FRACTIONS 


Example  4. — Subtract  9f  from  17f . 

17|  =  17H=16ii+-H=16H 


Qi.  =      qi2  _      012. 


7H   Ans. 


Fig.  14. 

In  Example  3,  f  cannot  be  subtracted  from  f.  To 
increase  f  so  that  the  subtraction  is  possible  borrow  one 
from  the  whole  number  5,  leaving  the  whole  number  4, 
change  1  to  eighths  (1  =  f ),  and  add  to  |.  The  upper  number 
becomes  4-V-,  from  which  If  can  be  subtracted. 

In  Example  4,  the  same  plan  is  used,  except  that  after 
1  is  borrowed  from  17,  it  is  changed  to  fifteenths.  In  each 
problem  change  the  1  borrowed  to  a  fraction  having  the 
L.  C.  D.  for  numerator  and  denominator. 

Observe  that  the  correctness  of  the  answer  can  be  checked 
by  going  over  the  solution  carefully,  or  by  adding  the  answer 
to  the  number  subtracted,  as  suggested  in  Example  3. 
The  student  should  form  the  habit  of  checking  each  solution. 


SUBTRACTION 


13 


^ 


Subtract: 

1.  Ill 

21 


EXERCISE  6 


2.  4i 


3.  11t^ 
41 


5.  15i-6T^  =  ? 

6.  14   -511=  ? 

7.  8i+7i-3i  =  ? 

8.  9|-2|  +  li  =  ? 

9.  Find  the  missing  dimension,  Fig.  15. 


4.  83i 


Ans.    8i 


7 

^ 

.  23          ->J 

1 

"'       1 

-lef 


Fig.  15. 
10.  Find  the  missing  dimension,  Fig.  16. 


Fig.  16. 


14 


FRACTIONS 


11. 


-1^ 


l'-^->\<- 


Find  the 
missing  di- 
mension, 
Fig.  17. 


-?.i- 


->H- 


-7f- 


Fig.  17. 
12.  Find  the  missing  dimension,  Fig.  18. 


Fig.  18. 
13.  Find  the  lengths  indicated  in  Fig.  19  (a),  (b)  (c)  and 
(d),  and  check  by  measuring. 

'\-^ 1 H  "h — ?-H 


1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 


1  II  I  I  M 


(a) 


MM  M 


ITT 


ir) 


Fig.  19. 


MULTIPLICATION  15 

8.  Multiplication  of  Common  Fractions. — The  method  of 
multiplying  one  fraction  by  another  or  by  a  whole  or  mixed 
number  is  shown  in  the  following  examples: 

Example  1.— Multiply  f  by  f .     f  X  f  =  ii- 
Example  2.— Multiply  f  by  9. 

Example  3.— Multiply  2|  by  f. 

^sX^-^X^     14     ^T¥- 
2 

Example  4.— Multiply  ^  by  4. 

3fX4  =  -V-XT  =  ^=15f. 
Example  5.— Multiply  4f  by  1^. 

12      7 
';4     P,o     ft4 
45X134-  ^X^^     17       ^• 
17 
Example  6. — Square  3^. 


oiv'^i  —  i:^Vi:^  —  169  —  10' 


Rule. —  To  multiply  fractions,  cancel  the  common  factors; 
the  required  result  will  he  the  product  of  the  remaining  numer- 
ators over  the  product  of  the  remaining  denominators. 

Observe  that  in  Example  2,  the  whole  number  9  is 
expressed  as  a  fraction  with  1  as  the  denominator. 

In  problem  3  the  mixed  number,  2|,  is  changed  to  an 
improper  fraction,  %^-. 


16 


FRACTIONS 


In  problem  6,  to  square  any  number,  multiply  it  by 
itself. 

EXERCISE  7 

1.  Multiply  i  by  f . 

2.  Multiply  t  by  StV- 

3.  Multiply  3|  by  6i 

4.  Multiply  f  X  2f  X  Y  X  |. 

5.  Find  the  product  of  9  and  144 1. 

6.  ^X7|X6ixH  =  ? 

7.  Find  the  product  of  A  and  7.     . 

8.  l|x4iXi  =  ? 

9.  Mensuration. — The  area  of  a  rectangle  equals  the 
product  of  the  base  and  the  altitude. 

The  perimeter  is  the  distance  around  a  figure. 

EXERCISE  8 
1.  Find  the  area  and    perimeter   of    the    rectangle  of 
Fig.  20. 


-6^ 


— base- 
Fig.  20. 


AREAS 


17 


2. 


-Hi"- 


Find  the  area  of  the  rectangle 
of  Fig.  21. 


Fig.  21. 


3.  Find  the  area  and  perimeter  of  Fig.  22, 

Ans.  28|  sq.  in. 
23^  in. 


51 


-=r— 2— -^ 


J 


i- 


Fig.  22. 
The  area  of  a  square  is  equal  to  the  square  of  one  side. 
EXERCISE  9 


Find  the  area  of  the  square, 
Fig.  23. 

Ans.  IQeT  sq.  in. 


Fig.  23. 


18 


FRACTIONS 


2. 


~^         Find  the  area  and  perimeter 
of  Fig.  24. 

Ans.  7i^  sq.  in.;   11  in. 


O 


.^ 


Find  the  area  of  the 
shaded  part  of  the  square, 
Fig.  25. 

Ans.  15tV  sq.  in. 
Fig.  25. 
4.  Measure  to  the  nearest  ye  inch  the  sides  of  the  two 
rectangles  in  Fig.  26  and  find  the  area  of  the  shaded  portion. 


Fig.  26. 


Measure  to  the  nearest 
YE  inch  the  sides  of  the 
two  squares  in  Fig.  27  and 
find  the  area  of  the  shaded 
portion. 
^^ 

Fig.  27. 


AREAS 


19 


The  area  of  a  triangle  equals  one-half  the  product  of  the 
base  and  altitude. 


EXERCISE  10 


1. 


Find  the    area  of 
~      the  triangle  shown  in 
Fig.  28. 

Ans.  9f  sq.  in. 


Fig.  28. 


2. 


Find    the    area    of 
Fig.  29. 

Ans.  12if  sq.  in. 


Fig.  29. 


3.  Find  the  area  of  the  triangle  in  Fig.  30. 

Ans.  16irr  sq.  in. 


Fig.  30. 


20 


FRACTIONS 


10.  Division  of  Fractions  and  Mixed  Numbers. — 
Example  1. — Divide  4^  by  4f . 


^2  ^^     12  •   8      U^f>     15- 


Rule. —  To  divide  fractions,  invert  the  divisor  and  follow 
exactly  the  rule  for  multiplication  of  fractions. 


1. 

2. 

3. 

4. 

5. 

11. 

12. 

13. 

weight. 


EXERCISE  11 

l-i-? 

6.  32-^3f  =  ? 

K4  =  ? 

7.  9f-^8|  =  ? 

8i^-l0  =  ? 

8.  6f^lf=? 

20^21  =  ? 

9.  16f^2i  =  ? 

f^6  =  ? 

10.  86f-^4|=? 

How  many 

eighths  in 

3?      (3-^|  =  ?) 

Divide  101 

into  56f . 

Six  castings 

5  weigh  118 1  pounds.     Fint 

Find  the  average 


14. 


Fig.  31. 


Find  the  base  of  the 
rectangle.  Fig.  31. 

Ans.  27i". 


Note. — We  have  learned  that  the  area  of  a  rectangle  =  base  X alti- 
tude. Therefore,  base  =  area  divided  by  altitude,  and  altitude  =  area 
divided  by  base. 


15.   r 


16. 


Fig.  32. 


21 

Find  the  base  of  the 
rectangle,  Fig.  32. 

oi  _ 


Find   the  altitude  of 

Fig.  33. 

Ans.  9i". 


Fig.  33. 


Fig.  34. 


Find   the  altitude  of 
Fig.  34. 


Note— The  area  of  a  triangle  =  ^  base X altitude.  Therefore,  the 
base  of  a  triangle  =  2  Xarea  divided  by  the  altitude,  and  the  altitude 
=  2Xarea  divided  by  the  base. 


18.  Find  the  base  of  the  triangle,  Fig.  35. 


Ans.  12f ". 


Fig.  35. 


22 


FRACTIONS 


19. 


Find  the  base  of 
the  triangle  of  Fig.  36, 
Ans.  IV'. 


Fig.  36. 
20.  Find  the  altitude  of  the  triangle  of  Fig.  37. 


Ans.  6' 


Fig.  37. 
21.  Find  the  altitude  of  the  triangle  of  Fig.  38. 


Ans.  51''. 


Fig.  38. 


APPLIED  PROBLEMS 


23 


22. 


Find  the  altitude  of 
the  triangle  of  Fig.  39. 
Ans.  12". 


Fig.  39. 

EXERCISE  12 
Applied  Problems.     Common  Fractions. 
1.  Find  the  diameter  of  the  smaller  circle,  Fig.  40. 


Fig.  40. 


24 


FRACTIONS 


3. 


Find  the  diameter 
of  each  circle,  Fig.  41. 


Find  the  thickness  cf 
the  ring,  Fig.  42. 

Outer  diameter  2f ". 
Inner  diameter,  2tg". 


Fig.  42. 
4.  Find  the  distance  between  centers,  Fig.  43. 


Fig.  43. 


THREADS  25 

Diameter  of  large  circle  is  5|". 
Diameter  of  small  circle  is  3]". 

5.  A  mechanic  earns  $6.00  in  8^  hours.     What  does 
he  earn  per  hour? 

6.  Find  the  length  of  the  rod  from  which  15  pieces, 
each  4f "  in  length,  can  be  cut? 

7.  Which  is  the  larger,  f  or  xV?     How  much? 

8.  In  a  certain  drawing  |"  represents  1'.  What  length 
of  line  will  represent  14  feet?  Ans.   If". 

9.  If  the  scale  is  -^"  =  V  Q",  what  distance  in  the 
drawing  represents  200  feet?  Ans.   12|". 

10.  If  a  planer  cuts  a  strip  -^'^  wide,  how  many  cuts 
are  necessary  to  plane  a  piece  3^"  wide?  Ans.   50. 

Note. — The  last  cut  may  not  be  as  wide,  but  it  is  the  same  length 
as  the  other  cuts. 

11.  If  a  sheet  of  metal  is  y&"  thick,  how  many  sheets 
are  required  to  make  a  pile  one  foot  thick?         Ans.   28. 

12.  An  armature  core  is  built  up  of  sheets  of  steel  -^ 
inch  thick.  How  many  sheets  are  required  to  build  an 
armature  9|  inches  thick?  Ans.  52. 

11.  Lead  and  Pitch  of  Screw  Threads. — The  distance 
from  any  point  on  a  screw  thread  to  the  corresponding 
point  on  the  next  thread  is  the  Pitch  of  the  thread  (see 
Fig.  44). 

Rule.  To  find  the  pitch  v)hen  the  number  of  threads  per 
inch  is  given,  divide  1  inch  by  the  number  of  threads  per  inch. 


26 


FRACTIONS 


In  Fig  44,  there   are  six  threads  per  inch;    l"-^6  =  i", 
the  pitch. 


ABC 
Fig.  44.— Single  V  Thread. 

If  the  thread  of  Fig.  44  is  traced  around  the  screw 
from  A,  it  will  reappear  at  B,  and  on  the  next  turn  at  C. 
The  so-called  threads  of  this  screw  are  parts  of  one  thread, 
or  it  is  single-threaded.  If  the  screw  is  turned  in  a  nut,  it 
will  advance  through  the  nut  |  of  an  inch  in  each  revolu- 
tion.    This  advance  is  called  the  Lead. 

On  a  single-threaded  screw  the  pitch  and  lead  are  equal. 

In  Fig.    45,  if  the  thread  starting  at  A  is  traced  around 


A     B     C     D 
Fig.  45.— Double  V  Thread. 


THREADS  27 

the  screw,  it  reappears  at  C;  starting  at  B,  it  reappears 
at  D.  Evidently  there  are  two  threads  on  the  screw,  or 
it  is  double-threaded. 

The  lead  on  a  double-threaded  screiv  is  twice  the  pitch. 

Similarly  a  screw  may  be  triple-threaded,  four-threaded, 
or  more.  To  obtain  the  lead  of  such  a  screw,  multiply 
the  pitch  by  3,  4,  etc. 

The  feed  of  a  lathe  is  the  distance  the  cutting  tool 
advances  for  each  revolution  of  the  stock. 

The  feed  of  a  drill  is  the  depth  of  cut  in  each  revolu- 
tion of  the  drill. 

EXERCISE  13.     APPLIED    PROBLEMS 

X.  What  is  the  pitch  of  a  single-threaded  screw  that  has 
8  threads  to  the  inch?  10  threads?  2f  threads?  What  is 
the  lead  for  each  screw? 

2.  Find  the  pitch  and  lead  of  double-threaded  screws 
having  3,  5|,  and  12  threads  to  the  inch,  respectively. 

3.  How  many  threads  per  inch  on  a  screw  if  the  pitch 
is  I  of  an  inch?  If  f  of  an  inch?  {Suggestion:  Divide  1 
inch  by  the  pitch.) 

4.  A  drill  cuts  1  inch  in  40  revolutions.     What  is  the  feed? 

5.  A  drill  will  cut  through  a  certain  steel  plate  1  inch 
thick  in  64  revolutions.  What  is  the  feed?  How  many 
revolutions  are  necessary  to  drill  5  holes? 

6.  How  many  minutes  will  be  required  to  turn  a  stock 
3  J  feet  long,  if  the  feed  is  -gV",  and  60  revolutions  are  made 
per  minute?  Ans.  20f. 

7.  A  piece  56  inches  in  length,  making  80  revolutions 
per  minute,  was  turned  in  35  minutes.     What  was  the  feed? 

Ans.  sV". 


28  FRACTIONS 

8.  Find  the  time  required  for  drilling  8  holes  in  a  steel 
plate  W  thick,  feed  ^",  R.  P.  M.  120.  Allow  2^  minutes 
per  hole  for  adjusting  the  machine. 

Ans.  22x1  minutes. 

12.  Gears. — A  wheel  with  teeth  on  its  rim  is  a  gear. 
When  gears  are  in  mesh,  the  number  of  revolutions  of  the 
driven  gear  is  found  by  dividing  the  product  of  revolutions 
of  the  driving  gear  and  the  number  of  its  teeth  by  the 
number  of  the    teeth  of  the  driven  gear. 

The  above  rule  may  be  expressed  in  this  manner: 
Revolutions  of  the  driven  gear  = 

Revolutions  of  the  driving  gearX  Teeth  of  driving  gear 
Teeth  of  driven  gear 
Or  when:  7^  =  Number  of  revolutions  of  driving  gear. 
r  =  Number  of  revolutions  of  driven  gear. 
!r=  Number  of  teeth  of  driving  gear. 
^  =  Number  of  teeth  of  driven  gear. 
The  above  statement  becomes, 

RXT 


r= 


t 


,    RXT        _,     rXt  T_r><t 

also,  t  =  ——  ;      K  —  -^ ;  i  —   ^  . 

EXERCISE  14.  GEARS 
Find  the  missing  values : 

1.  R=l  T  =  27  t  =  9  r  =  ? 
R  =  2  T=18  t  =  ?  r  =  6 
R  =  ?  T  =  24  t  =  Q  r  =  7 
R  =  3            T  =  ?  ^=12  r  =  3^ 

2.  A   16-tooth  gear  is  driving  a   12-tooth  gear.     How 
many  revolutions  will  the  driven  gear  make  to  one  of  the 


GEARS 


29 


driving  gear.     How  many  revolutions  will  the  driving  gear 
make  to  one  of  the  driven  gear? 

Ans.  If     !• 
3.  In  Fig.  46,  how  many  revolutions   does  the   clutch 


MAIN  DRIVE  SHAFT 


") 


CLUTCH  SHAFT 
COUNTER  SHAFT 


Fig.  46.^First,  or  "  low  "  speed;   gears  turn  in  direction  of  arrows. 

shaft  make  to  one  revolution  of  the  main  drive  shaft.  Sug- 
gestion: Consider  gear  4  to  make  one  revolution;  find 
number  of  revolutions  of  gear  1.  Ans.   4|-  or  4.4. 


33  Tecthxv^5 


25  Teeth       ,     „       ^ 
/  16  Teeth 

■-     'ifl^^        III  Fig-  4"^  i^  shown 

the    meshing   of    gears 

in   intermediate   speed. 

Find  the  gear  ratio. 

^¥~    Ans.    22*3,  or  nearly  2.7 

to  1. 


Fig.  47. — Second,  or  "  intermediate  " 
speed;   gears  turn  in  direction  of  arrows. 


CHAPTER  II 
DECIMAL  FRACTIONS 

13.  Definition. — Fractions  that  have  10  or  some  power 
of  10  for  a  denominator  are  decimal  fractions. 

For  example,  to,  to¥,  toooo,  are  decimal  fractions. 
These  fractions  may  be  written,  .1,  .07,  .0145,  the  decimal 
point  being  used  in  the  place  of  the  denominator.  The 
last  figure  of  each  numerator  is  written  as  many  places  to 
the  right  of  the  decimal  point  as  there  are  ciphers  in  the 
denominator  of  the  common  fraction. 

14.  Reading  Decimal  Fractions.— i^  decimal  fraction  is 
read  the  same  as  the  corresponding  common  fraction. 
Thus,  .1,  .02,  .027,  are  read  as  if  written  to,  too  rior- 
When  a  whole  number  and  a  decimal  are  written  togethei 
as  6.17,  the  number  is  read  as  if  written  6iW,  or  six  and 
seventeen  hundredths,  the  word  "  and  "  indicating  the  posi- 
tion of  the  decimal  point. 

Note. — Very  often  the  above  fraction  is  read,  "  six  point  one  seven." 
The  following  is  given  as  an  aid  in  reading  decimals: 


-c 

M 

c 

r^ 

03 

« 

-t-i 

m 

■e 

X3 

3 

•i 

m 

C 

O 

w 

r** 

i3 

^ 

(S 

-t-i 

3 

r-i 

,__, 

V. 

r-> 

O 

T3 

-2 
•a 

c3 

a 

(V 

3 

00 
O 

.2 

1 

p 

n 

H 

K 

H 

H 

ffi 

000  .000000 

30 


ADDITION  AND  SUBTRACTION  31 

EXERCISE  1 
Read  the  following  decimals: 

1.  .0076  3.     23.9026  5.  1674.01674 

2.  7.986  4.  275.275  6.       27.0467 

Write  the  following  in  figures: 

7.  Eight  hundred  ninety-five  ten  thousandths. 

8.  Two  hundred  ninety  seven  hundred  thousandths. 

9.  Four  hundred  thirty-one  and  sixty-two  thousandths. 
10.  One  hundred  and  twenty-seven  millionths. 

15.  Addition  of  Decimals : 

Example.— Add  6.749,  23.0764,  .00072,  100.0000702. 

6.749 

23.0764 

.00072 
100.0000702 


129.8261902 


Rule. — Place  the  numbers  to  be  added  so  that  the  decimal 
points  form,  a  straight  line,  and  proceed  as  in  the  addition 
of  whole  numbers. 

16.  Subtraction  of  Decimals : 

Example.— Subtract  10.0846  from  15.00001. 
15.00001 
10.08460 

4.91541 


32  DECIMAL  FRACTIONS 

17.  Multiplication  of  Decimals : 
Example.— Multiply  1003.21  by  .064. 
1003.21 
.064 


401284 
601926 

64.20544 


Rule. — Multiply  as  in  whole  numbers  and  point  off  as 
many  decimal  places  in  the  product  as  there  are  in  the  factors 
taken  together. 

EXERCISE  2 

1.  Add  3.985,  4.06,  98.4763,  .0356,  and  100.728. 

2.  Add  13.8505,  909.73,  .00007,  and  148.561. 

3.  Add  978.001,  100.999,  47.0878,  and  79.38. 

4.  Add  59.86,  .13,  9.9,  .008,  and  749. 

5.  Subtract  19.496  from  97.382. 

6.  Subtract  .987634  from  1.00175. 

7.  Subtract  123.4567  from  987.8321. 

8.  Subtract  .3594  from  1. 

Find  the  product  in  the  following: 

9.  3.945X8.6. 

10.  578.01X49.7. 

11.  .003X7.8. 

12.  79.05 X. 46. 

13.  98.8 X. 069. 

14.  3.1416X.25. 

15.  6.321X100. 


DIVISION  33 

18.  Division  of  Decimals. — -Placing  the  decimal  point 
in  the  quotient  is  the  greatest  difficulty  the  student  has  in 
mastering  division  of  decimal  fractions.  A  careful  study 
of  the  following  examples  will  aid  the  student  in  overcoming 
the  difficulty. 

Example  1.— Divide  .0093  by  4. 

■  002325  Quotient 
Divisor  4). 009300  Dividend 

Divide  as  in  whole  numbers,  annexing  ciphers  if  necessary 
(two  were  annexed  in  this  problem).  The  decimal  point 
of  the  quotient  is  placed  directly  above  the  decimal  point 
of  the  dividend;  when  the  divisor  is  a  whole  number  there 
are  as  many  decimal  places  in  the  answer  as  in  the  dividend. 
In  the  discussion  of  common  fractions  it  was  shown 
that  both  numerator  and  denominator  of  a  fraction  may 
be  multiplied  or  divided  by  the  same  number  without 
changing  the  value  of  the  fraction. 

TVinc  3. 6    _15_30_3000      „J.p 

J-ilLlb  8  ~  1  6  —  40  —  80  —  8000J   t^l^*^' 

Example  2.— Divide  3354  by  .078. 

3354 

This  problem  may  be  written  as  a  fraction,  .     If 

the  numerator  and  denominator  are  multiplied  by  1000,  it 
gives, 

3354     1000    3354000 

.078^1000"      78      ' 

this  fraction  can  be  reduced  by  division.     Ordinarily  the 
problem  is  solved  as  follows: 


34  DECIMAL  FRACTIONS 

43000.  Quotient 
Divisor  .078x) 3354.000^  Dividend 
312 

234 
234 

000 

If  the  decimal  points  of  the  dividend  and  the  divisor  are 
shifted  three  places  to  the  right,  the  result  is  the  same  as 
was  obtained  by  multiplying  both  by  1000.  The  new 
positions  of  the  decimal  points  are  indicated  by  x- 

Place  the  first  figure  of  the  quotient  above  the  last  figure  of 
the  first  product;  4  in  the  quotient  is  written  above  2  in  312. 

Example  3.— Divide X383. 594  by  48.7. 

14.0368  Quotient 


Divisor  48. 

7x) 683. 5x9400  Dividend 

487 

1965 

1948 

1794 

1461 

3330 

2922 

4080 

3896 

184  Remainder, 

DIVISION  35 

When  the  dividend  and  the  divisor  are  multipUed  by  10, 
the  divisor  has  no  decimal  places.  Place  x  in  the  dividend 
and  divisor  one  place  to  the  right  of  the  decimal  point. 
The  first  figure  and  decimal  point  of  the  quotient  are  placed 
as  suggested  in  Example  2. 

When  the  third  subtraction  is  made  there  is  a  remainder, 
333.  Further  division  is  possible  if  one  or  more  ciphers 
are  annexed  to  the  dividend.  The  final  remainder  indi- 
cates that  the  quotient  obtained  is  not  exact. 

Note. — Pointing  off  to  the  iiearest  thousandth  :  Since  .0368  is 
nearer  to  .0370  than  to  .0360  in  value,  the  quotient  may  be  written 
14.037.  Division  to  the  fourth  place  is  necessary  to  determine  this 
result. 

EXERCISE  3 
Divide  the  following: 

1.  .085^4  7.  5771.402-7.34 

2.  98.07^3  8.  5.698^154 

3.  .7656^8  9.  .01-^8 

4.  126^.07  10.  1.296-^1.8 
^.  1833^3.9  11.  3.4592^73.6 

6.  4.56^.16  12.  22.002 H- .057 

13.  Divide  to  the  nearest  .001. 

70.90742^14.6 

14.  Divide  28.52806  by  8.6. 

15.  Divide  $90  by  24. 

19.  Reduction  of  Common  Fractions  to  Decimals. — A 

common  fraction  may  be  reduced  to  a  decimal  by  dividing 
the  numerator  by  the  denominator. 


36  DECIMAL  FRACTIONS 


Example 

1.- 

— Reduce  xe  to  a  decimal, 
1 . 1875 
16)3.0000 
1  6 

1  40 

1  28 

120 
112 

80 
80 

Example  2. — Reduce  rs  to  a  decimal. 
1.0666 


Ans.  .1875. 


15)1.0000  Ans.  .067. 

90 


100 
90 

100 
90 

10 
FXERCISE  4 

1.  Reduce  the   following  fractions   to  decimals:  -^o?  tt> 

A      i       2.      _A_      5. 
7)     3)     9)     11)     6- 

2.  Divide  6f  by  Ig-  (use  fractions). 

3.  Change   the   fractions   in  problem  2  to  decimals  and 
divide. 


PERCENTAGE 


37 


Compare  the  two  answers. 

4.  Change  to  decimals  and  add: 

3.1 5 I 1 I 3 I       9      15 

8ll6~32r64ll6l8. 

Ans.  1.953125. 

5.  Decimal  equivalents  of  fractions,  ^  inch  to  1  inch; 
solve  for  the  missing  values. 


17 
64 
9 

32 
19 
64 


TS 


=  ? 


A  =.078125 
A  =.09375 
6T=.  109375 

i  =  .1250 
^=.140625 
^f=.  15625 

i^=.1875 
if  =.203 125 

7  . 

T2- 

1  5  . 

ST- 
i=? 


.265625 

.28125 

.296875 

.3125 

.328125 

.34375 

.359375 


=  '? 


.390625 

.40625 

.421875 

.4375 

.453125 

.46875 

.484375 

.5000 


If  =  .515625 
H  =  .53125 
tf  =  . 546875 

_9__? 
16  —  • 

fi=. 578125 
if  =.59375 
If  =.609375 

I  =  .6250 
il=. 640625 

M=? 

M=. 671875 
ii=.6875 

6T~ 


f  =  .7500 


||  =  .  765625 
|f  =  ;78125 
If  =  .796875 


if  =.828 125 
M=. 84375 
tf=. 859375 
I  =  .8750 

64  —  ■ 

ft  =.90625 


lt=.9375 
If  =.953 125 

3J._  9 
32  —  • 

If  =.984375 


1  =  1 


20.  Percentage. — Percent  means  by  the  hundred,  and  is 
written  "  %." 

Examples.— 50  per  cent  of  72  =  t%°oX72,  =.50X72  =  36. 
12%of  45  =  tWX45=. 12X45  =  5.4. 

Observe  that  the  symbol  "  %  "  is  used  instead  of  the 
denominator  100,  or  two  decimal  places.  In  the  solution 
of  problems,  the  decimal  form  is  used  most  frequently. 


ar.^!ir^r>^ 


38  DECIMAL  FRACTIONS 

EXERCISE  5 

1.  Find  15%  of  60;  87%  of  387;  37|%  of  472  ft;  33|% 
of  $72;  i%  (i  of  1%)  of  640";  |%  of  1728  cubic  inches; 
7^%  of  1585  pounds;  125%  of  48;  12^%  of  48;  682%  of 
956. 

2.  Aluminum  bronze  is  10%  aluminum  and  90%  copper. 
How  many  pounds  of  each  metal  in  125  pounds  of  the 
alloy? 

Ans.  12i  lbs.,  112^  lbs. 

3.  If  a  lathe  costing  $1350  depreciates  (decreases  in 
value),  12^%  in  one  year,  what  is  it  worth  at  the  end  of  the 
year? 

Ans.  $1181.25. 

4.  Bonds  of  the  Fourth  Liberty  Loan  bear  4j%  interest, 
payable  semiannually.  How  much  interest  is  due  April  15, 
and  October  15,  on  bonds  having  a  par  value  of  $650? 

Ans.  $13.81. 

21.  Relation  of  Numbers  Expressed  as  a  Percent. — To 

express  one  number  as  a  percent  of  another  numl^er,  divide 
the  first  by  the  second  to  the  nearest  .01,  and  write  the 
result  as  percent. 


Examples. — 


20  —  .15—  15%; 
-V-- 5.00  =  500%; 


f=. 2857  =  28.57%. 

Note. — Where  greater  accuracy  is  desired,   decimal    places  after 
the  second  may  be  retained  in  decimal  form  as  in  the  last  example. 


EFFICIENCY  39 


EXERCISE  6 


1.  A  gear  blank  in  the  rough  weighed  9  pounds.  The 
finished  piece  weighed  7  pounds.  How  many  pounds  were 
wasted?     What  percent  of  the  blank  was  wasted? 

Ans.  22%. 

2.  In  250  pounds  of  type  metal  there  are  200  pounds  of 
lead  and  50  pounds  of  antimony.  What  is  the  percentage 
of  each  metal? 

Ans.  80%,  20%. 

3.  Gold  coin  is  900  parts  gold,  75  parts  copper,  and  25 
parts  silver.  What  is  the  percentage  of  each  metal  in  the 
alloy?    Suggestion:  What  is  the  total  number  of  parts? 

Ans.  90%,  7i%,  2i%. 

4.  An  article  which  cost  $375  was  sold  for  $425.  The 
profit  was  what  percent  of  the  cost? 

Ans.  13.3%. 

22.  Efficiency. — The  efficiency  of  a  machine  is  the  ratio 
of  useful  work  the  machine  does  to  the  energy  it  receives. 
It  may  be  stated  thus : 

Output 


Efficiency  = 


Input  * 


If  a  power  plant  converted  all  the  fuel  into  useful  power, 
output  would  equal  input,  and  the  power  plant  efficiency 
would  be  100%.  If  a  steam  engine,  using  fuel  that 
contains  100  horse  power  of  energy,  delivers  12  horse 
power  to  the  belt  or  drive  wheel,  its  thermal  efficiency  is 
T^  or  12%. 


40 


DECIMAL  FRACTIONS 


EXERCISE  7 

Compute  the  efficiency  of  the  following: 

INPUT.  OUTPUT.  EFFICIENCY. 

Gasoline  Motor 150  66  ? 

Diesel  Engine 200  154  ? 

Electric  Motor 45  37.4  ? 

23.  The  Micrometer. — -The  micrometer  is  used  to 
measm'e  to  one  thousandth  of  an  inch.  The  parts  are 
named  as  shown  in  Fig.  48.     The  anvil  and  sleeve  are  attached 


Set  Screw  Spindle  of  Screw 

\  Anvil       / 


Ratchet 


Thimble 


Frame  of  Yoke 

Fig.  48. 


to  the  frame,  and  are  not  movable.  The  sjyindle  turns 
in  the  sleeve.  It  is  a  single  threaded  screw  with  40  threads 
to  the  inch.  Since  the  pitch  of  the  thread  is  tV,  or  .025 
in.,  one  turn  of  the  screw  changes  the  width  of  the  opening 
.025  in.  This  is  the  distance  the  thimble  moves  on  the 
sleeve;  and  its  position  at  the  end  of  each  revolution  is 
indicated  by  the  marks  on  the  sleeve.  Every  fourth  mark 
is  longer;  the  distance  between  these  marks  is  4X.025  in. 
or  .100  in. 


THE  MICROMETER 


41 


The  thimble  turns  with  the  spindle.  The  beveled  edge 
has  25  divisions.  When  the  thimble  is  turned  2V  of  one 
revolution,  the  screw  moves  -it  of  .025  in.  or  .001  in. 

There  are  four  steps  in  reading  the  micrometer: 

1.  Read  the  tenths  of  an  inch  from  the  numbered  marks 
on  the  sleeve. 

2.  Multiply  the  number  of  divisions  to  the  right  of  the 
last  numbered  mark  on  the  sleeve  by  .025. 

3.  Read  the  thousandths  on  the  thimble. 

4.  Add  the  three  results. 


/ 

/ 

\ 

■10 

yo     1    2 

.     h 

X 

W 

Fig.  49. 


4  5  6 


Fig.  51. 


42  DECIMAL  FRACTIONS 

In  Fig.  49,  the  reading  .257,  is  obtained  as  follows: 
2 X. 100  =  .200  in. 
2 X. 025  =  .050  in. 
7X. 001  =  .007  in. 


.257  in. 


EXERCISE  8 

1.  What  is  the  reading  in  Fig.  50? 

2.  What  is  the  reading  in  Fig.  51? 

3.  What  is  the  reading  in  Fig.  52? 

EXERCISE  9.     REVIEW  PROBLEMS 

1.  Tin  weighs  459  pounds  per  cubic  foot.    "What  is  the 

weight   of   one   cubic   inch?     (1728   cubic   inches  =1    cubic 

■foot.) 

Ans.  .2656  lb. 

2.  If  cast  iron  weighs  .261  pound  per  cubic  inch,  how 
many  cubic  inches  in  an  iron  casting  weighing  845  pounds? 

Ans.  3237.5  cubic  inches. 
How  much  more  would  the  same  volume  of  steel  weigh? 
(Steel  weighs  .283  pound  per  cubic  inch.) 

Ans.  71.22  pounds. 

3.  What  is  the  cost  of  1250  feet  of  wrought  iron,  weigh- 
ing 1.83  pounds  per  foot,  and  costing  95  dollars  a  ton? 

Ans.  $108.65. 

4.  The  weight  of  one  cubic  foot  of  water  is  62.3  pounds, 
and  the  weight  of  the  same  volume  of  rolled  brass  is  524 
pounds.  How  many  times  as  heavy  as  water  is  rolled 
brass?     How  many  times  as  heavy  as  rolled  brass  is  water? 

Ans.  8.41;  .1189. 


TAPER  43 

5.  Number  6,  U.  S.  Standard  Plate  Iron  and  Steel  is 
.203125  inch  thick  and  weighs  about  4.96  pounds  per 
square  foot.  How  many  sheets  are  in  a  stack  6.5  inches 
high?  What  is  the  weight  of  this  stack  if  each  sheet  con- 
tains 9  square  feet? 

Ans.  32  sheets;    1428.48  pounds. 

24.  Taper — Taper  is  the  decrease  in  the  diameter  of  a 
piece  of  work  in  a  unit  of  length,  usually  one  foot.  In  some 
problems  the  taper  per  inch  is  given ;  in  others  it  is  convenient 


Fig.  53. 

to  find  the  taper  per  inch.  In  Fig.  53  the  taper  per  foot 
is  I  in.  (1"  — I")  or  .500  in.  Distance  between  centers 
rather  than  the  actual  length  of  the  stock  must  be  consid- 
ered in  computing  taper.  In  the  shop,  detailed  instructions 
are  given  for  adjusting  machines  for  taper  cutting. 

EXERCISE  10 

1.  The  diameters  of  the  ends  of  a  piece  one  foot  long 
are  1  inch  and  .400  inch,  respectively.     What  is  the  taper? 

2.  Find  the  taper  of  the  finished  piece  of  work  6  inches 
in  length,  if  the  diameter  of  the  large  end  is  1.5  inches  and 
of  the  smaller  end  1.25  inches.  Ans.  .5  in.  per  foot. 

3.  The  taper  of  a  certain  piece  of  work  7  inches  long 
is  .600  inch  per  foot.  The  diameter  of  the  large  end  is 
1|  inches.     Find  the  diameter  of  the  small  end? 

Ans.  .775  inch. 


44 


DECIMAL  FRACTIONS 


25.  The    Circle. 


Fig.  54. 


The  radius  of  a  circle  is  one-half  of 
the  diameter.  The  circumference  is 
3.1416  times  the  diameter. 

This  rule  may  be  stated  as  a 
formula, 

C  =  tD,     or     C  =  2wR. 

The  letter  w,  called  "  pi,"  has 
the  value  3.1416.  If  C  is  given, 
D  and  R  may  be  found  by  using  the 
formulas; 


D  = 


C 


-£ 


EXERCISE  11 

1.  (a)  Find  C  when  /)  =  7.15  inches. 
(b)  when  /?  =  4.54  inches. 

Ans.  (a)  22.46244  inches;   (h)  28.5257  inches. 

2.  Find  D  and  R  when  C=  11.781. 

Ans.  D  =  3.75  inches;  /?=  1.875  inches. 

3.  A  fly  wheel  is  42  inches  in  diameter,  and  makes  250 
R.  P.  M.  At  what  speed  (feet  per  minute)  does  it  drive  a 
belt? 

Ans.  2748.9  feet. 

26.  Cutting  Speed. — The  rate  at  which  a  point  on  the 
surface  of  a  piece  of  stock  turns  past  the  tool  in  a  lathe 
is  the  Cutting  Speed.  The  distance  traveled  in  one  revolu- 
tion is  the  circumference  of  a  circle  having  a  diameter  equal 
to  the  diameter  of  the  piece  of  stock.  This  distance  multi- 
plied by  the  number  of  revolutions  per  minute  (R.P.M.) 


CUTTING  SPEED  45 

gives  the  Cutting  Speed.     The  answer  should  be  expressed 
in  feet. 

Example   1. — ^Find    the  cutting  speed  when    a  l|-inch 
rod  is  revolving  95  times  per  minute. 

1^X3.1416  =  4.7124,  the  circumference  in  inches. 

4.7124X95 


12 


=  37.3,  cutting  speed  in  feet. 


EXERCISE  12 


1.  What  is  the  cutting  speed  when  2|-inch  stock  makes 
175  R.P.M.? 

Ans.  97. 

2.  At  how  many  R.P.M.  should  If-inch  stock  be  turned 
in  a  lathe  to  give  a  cutting  speed  of  40  feet  per  minute? 

Ans.  87. 

27.  Areas. — The  area  of  a  circle  is  found  by  multiplying 
the  square  of  the  diameter  by  .7854. 

The  formula  is,  A  =  .7854^^-. 

Example. — Find  the  area  of  a  circle  whose  diameter  is 
5  inches. 

A  =  5X5X. 7854=  19.635,  area  in  square  inches. 

EXERCISE  13 

1.  The  diameter  of  the  piston  of  a  gas  engine  is  3^ 
inches.      Find  the  area.  Ans.  9.621  square  inches. 

2.  The  diameter  of  a  circle  is  one  inch.    Find  the  area. 

Ans.  .7854  square  inch. 

3.  The  circumference  of  a  circle  is  14.1372  inches.  What 
is  the  area? 

Ans.  15.9  square  inches. 


46 


DECIMAL  FRACTIONS 


28.  Volumes.— jT/ie  volume  of  a  rectangular  solid  or 
cylinder  is  equal  to  the  product  of  the  area  of  he  base  and  the 
altitude. 


Fig.  55. 


Fig.  Sf). 


Fig.  58. 


In  Fig.  55,  8"X5"  =  40  square  inches,  area  of  base. 
40  square  inchesXl2"=480  cubic  inches,  volume  of  solid. 

In  Fig.  57  or  58,  3"  X 3"  X 3. 1416  =  28.2744  square  inches, 
area  of  base.  28.2744  square  inches  X  12"  =  339.292  cubic 
inches,  volume  of  solid. 


VOLUME  OF  SOLIDS  47 

EXERCISE  14 

1.  Find  the  volume  of  the  sohd  shown  in  Fig.  56. 

2.  Find  the  volume  of  a  rectangular  solid  of  steel  whose 
edges  are  6  inches,  5  inches,  and  3  inches  respectively.  What 
is  its  weight  if  steel  weighs  490  pounds  per  cubic  foot? 

Ans.  25.52  pounds. 

3.  How  many  solids,  each  containing  .875  cubic  inch, 
can  be  taken  from  a  bar  of  steel  the  area  of  the  base  being 
six  square  inches,  and  the  length  12  feet?  Allow  18%  for 
waste. 

Ans.  809. 

4.  A  cylindrical  Portland  cement  pillar  is  12  inches  in 
diameter,  and  15  feet  high.  If  Portland  cement  weighs 
90  pounds  per  cubic  foot,  what  is  the  weight  of  the  pillar? 

Ans.  1060.29  pounds. 

5.  Find  the  weight  of  a  lead  pipe  10  feet  long,  if  it  weighs 
5.75  pounds  per  foot.  Compare  this  result  with  the  result 
obtained  when  the  pipe  is  considered  as  the  difference 
between  the  volumes  of  two  right  circular  cylinders,  whose 
lengths  are  10  feet  each,  and  whose  diameters  are  1.75  and 
1.25  inches,  respectively.  (Lead  weighs  .4105  pound  per 
cubic  inch.) 

Ans.  58.03  pounds. 


CHAPTER  III 
POWERS  AND  ROOTS 

29.  Powers. — If  a  number  is  used  two  or  more  times 
as  a  factor,  the  product  is  a  power  of  the  number.  4X4=16, 
the  second  power  or  square  of  4.  4X4X4  =  64,  the  third 
power  or  cube  of  4.  4X4X4X4  =  256,  the  fourth  power 
of  4. 

The  writing  of  the  factors  may  be  shortened  by  using 
the  common  factor  or  base  once,  and  putting  above  and  to 
the  right,  a  small  number,  showing  how  many  times  the 
base  is  used  as  a  factor.  This  number  is  an  Exponent. 
4^  means  4X4,  2  being  the  exponent.  Similarly,  5'^  means 
5X5X5;  9-i  means  9X9X9X9. 

EXERCISE  1 

1.  Find  the  value  of  6^,  15^,  8^  5^. 

2.  Square  3.  Compare  the  square  of  the  result  with 
the  fourth  power  of  3. 

3.  Which  is  the  larger,  6-*  or  IP? 

4.  Add  the  square  of  12  to  the  square  of  16.  Compare 
this  with  the  square  of  20. 

5.  What  is  the  value  of  (h)^  (2|)2,  (f)^? 

6.  Square  3.5;   11.25. 

7.  To  find  the  area  of  a  square,  square  the  length  of 
one  side.  Find  the  area  of  a  square  7^  inches  on  a  side. 
Of  one  16.2  inches  on  a  side.     Of  one  3|  inches  on  a  side. 

48 


SQUARE  ROOT  49 

30.  Roots. — A  root  of  a  number  is  one   of  the   equal 

factors  of  the  number.  Since  4X4=16,  4  is  a  root  of  16. 
Four  is  the  square  root  of  16,  because  4  squared  equals  16. 
Likewise,  since  4"^  =  64,  4  is  the  cube  root  of  64. 

The  radical  sign  V  is  placed  before  a  number  to  indicate 
that  a  root  of  the  number  is  to  be  found.  A  line  is  drawn 
over  the  number  to  show  how  much  of  what  follows  is 
affected  by  the  radical;  thus  V729.  If  there  is  no  number  in 
the  opening  of  the  radical  sign,  the  square  root  is  to  be  found. 

31.  Square  Root. — It  was  learned  in  the  multipli^tion 

table  that  3X3  =  9;    therefore,  V9  =  3 .     Similarly,  V25  =  5 ; 

V49  =  7. 

EXERCISE  2 

Find  by  reference  to  the  multiplication  table  or  by  experi- 
ment, the  square  root  of  the  following: 

1.  100         3.  1  5.  144         7.  16  9.  225 

2.  4  4.  64  6.  121  8.  400         10.  81 

32.  Computing  Square  Roots. — The  multiplication  table 
(facing  page  1)  does  not  include  products  larger  than  144  and 
only  a  few  of  the  products  in  the  table  are  squares.  For 
finding  the  square  roots  of  other  numbers  and  all  numbers 
larger  than  144,  another  method  is  commonly  used. 

Example. — Find  the  square  root  of  3844. 

6    2. 


38 '44. 
36 


122)244  Ans.  62. 

244 


50  POWERS  AND  ROOTS 

Beginning  at  the  decimal  point,  point  off  the  number 
into  periods  of  two  figures  each.  In  the  period  at  the  left, 
38,  find  the  largest  square.  This  is  36,  and  the  square 
root  of  36  is  6.  Write  6  above  the  first  period  and  write 
36  under  38.  Subtract  36  from  38  and  bring  down  the  next 
period  which  gives  244. 

Multiply  6  by  2,  writing  the  product,  12,  at  the  left. 
This  number,  12,  is  called  a  trial  divisor.  Divide  12  into 
24,  which  gives  the  quotient  2.  Annex  2  to  12,  making 
the  complete  divisor  122. 

Multiply  this  number  by  2  and  write  244  in  the  proper 
place.  Write  2  in  the  answer,  making  62  the  square  root 
of  the  number. 


Example  2. 


3    8.     0      7 
14'49.'32'49 
9 
68)540 
544 

7607  53249  Ans.  38.07. 

53249 


Point  off  as  in  Example  1,  beginning  at  the  decimal  point. 
Division  by  the  trial  divisor  6,  gives  9.  If  69  is  multi- 
plied by  9,  the  result  is  621,  which  is  larger  than  549.  Use 
8  as  the  last  figure  of  the  complete  divisor;  multiplying, 
the  result,  544,  which  can  be  subtracted  from  549,  is  obtained. 
In  the  next  step,  53  is  not  divisible  by  76.  In  such  case 
write  0  as  the  next  part  of  the  answer,  and  annex  0  to  the 


SQUARE  ROOT  51 

trial  divisor.  Then  bring  down  the  next  period.  Divide 
5324  by  760;  the  result  is  7,  the  last  figure  of  the  answer. 
Point  off  one  decimal  place,  counting  from  the  right,  in 
the  answer,  for  each  decimal  period  in  the  given  number. 

Example  3. — Find  the  square  root  of  764  to  the  nearest 
.001. 

2     7.     6     4     0     5 


7'64.'00'00'00' 

00 

4 

47) 

364 

3  29 

35  00 

546) 

32  76 
2  24  00 

5524) 

2  20  96 

3  04  00  00 

KK^<if) 

2  76  40  25 

K\                 _  

27  59  75 

Ans.  27.641. 


Annex  8  ciphers  after  the  decimal  point,  making  four 
periods;  the  root  must  be  found  to  four  decimal  places  to 
determine  the  nearest  .001.  The  solution  is  similar  to  that 
of  Examples  1  and  2.  There  is  a  remainder  which  indicates 
that  the  answer  is  not  exact.  When  the  fourth  decimal 
figure  of  the  answer  is  5  or  more,  increase  the  figure  in 
^hird  place  by  1.      .6405  to  the  nearest  .001  is  .641. 

To   find   the   square   root   of  a  fraction  find  the  square 


52  POWERS   AND   ROOTS 

root  of  numerator  and  denominator  separately,  and  write 
the  result  as  a  fraction. 

Example  4. — Find  the  square  root  of  Y^i. 

/441_  V441_21^7  .    g 

729~V729~27     9* 

If  the  denominator  is  not  a  perfect  square,  reduce  the 
fraction  to  an  equivalent  decimal,  and  find  the  square  root 
of  this  number. 

Example  5. — Find  the  square  root  of  ts  to  nearest  .001. 


V.  133338  =  .365     A 


ns. 


EXERCISE  3 

Find  the  square  root  of: 

1.  1024  Ans.  32  9.  tWt  Ans.  | 

2.  2916  54  10.  -iM  \ 

3.  361  19  11.  4W0  \ 

4.  7396  86  12.  %  .913 

5.  1,234,321  nil  13.  \  .7071 

6.  20.25  4.5  14.  2  1.4142 

7.  .000529  .023  15.  3  1.732 

8.  15239.9025  123.45  16.  5  2.236 

17.  Find  the  side  of  a  square  which  has  an  area  of  20. 

Ans.  4.472. 

18.  The  area  of  a  square  is  110.     What  is  the  length 

of  a  side? 

Ans.  10.488. 

19.  The  area  of  a  square  is  28879876.     Find  the  side. 

Ans.  5374. 


THE  RIGHT  TRIANGLE 


53 


20.  If  the  area  of  a  square  is  45.6976,  what  is  the  length 

of  a  side? 

Ans.  6.76. 

5 

33.  The  Right  Triangle. — A  right  triangle  has  a  right 
angle,  or  an  angle  of  90°.  The  two  sides  are  perpendicular 
to  each  other.  The  side  opposite  the  right  angle  is  called 
the  hypotenuse. 


Fig.  59. 

Problems  involving  the  right  triangle  require  a  knowledge 
of  squares  and  square  root  for  their  solution.  A  principle 
from  geometry  is  used  in  finding  the  length  of  one  line, 
when  the  other  two  are  given.     It  is  stated  thus: 

The  square  of  the  hypotenuse  is  equal  to  the  sum  of  the 
squares  of  the  other  two  sides. 

Example  1. — If  one  side  of  a  right  triangle  is  3,  and  the 
other  side  is  4,  find  the  hypotenuse. 

32  =  9      42=16       9  +  16  =  25 


\/25  =  5,  the  hypotenuse. 


54  POWERS  AND   ROOTS 

Example  2. — If  the  hypotenuse  is  10,  and  one  side  is 
6,  find  the  other  side. 

102=100       62  =  36       100-36  =  64 
V'64  =  8,  other  side. 

Observe  that  if  the  square  of  one  side  is  subtracted 
from  the  square  of  the  hypotenuse,  the  remainder  is  the 
square  of  the  other  side. 

EXERCISE  4 

1.  The  sides  of  a  right  triangle  are  16  and  30  inches, 
respectively,  find  the  hypotenuse. 

Ans.  34  in. 

2.  A  guy  wire  is  attached  to  a  wireless  tower  454  feet 
in  height,  and  is  anchored  724  feet  from  the  foot  of  the 
tower;  find  the  length  of  the  wire. 

Ans.  854.57  ft. 

3.  One  side  of  a  right  triangle  is  9  inches,  and  the  hypot- 
enuse is  41  inches.      Find  the  length  of  the  other  side. 

Ans.  40  in. 

4.  A  tree  breaks  off  10  feet  from  the  ground  and  the 
top  strikes  the  ground  40  feet  from  the  base  of  the  tree. 
How  tall  is  the  tree? 

Ans.  51.2  ft. 
Find  the  missing  part  in  each  problem  correct  to  .01. 

HYPOTENUSE.  SIDE.  SIDE.  ANS. 

5.  12  8  ?  8.94 

6.  ?  30  40  50 

7.  57  ?  19  53.74 

Note. — The  diagonal  of  a  square  or  rectangle  divides  the  figure 
into  two  equal  right  triangles. 


THE  RIGHT  TRIANGLE 


55 


8.  What  is  the  diagonal  of  a  square  10|  inches  on  a 
sivie?     Of  a  square  20  inches  on  a  side? 

Ans.  14.85  in;    28.284  in. 
0.  Measure  the  two  sides  of  the  triangle,  Fig.  60,  and 
find  the  third  side.     Check  by  measuring  the  third  side. 


Fig.  60. 


Fig.  61. 


10.  Measure  the  hypotenuse  and  one  side  of  the  triangle, 
Fig.  61,  and  find  the  other  side.  Check  by  measuring  the 
third  side. 

11. 


Find    the   diagonal   of    the 
rectangle  of  Fig.  62. 

Ans.  30.412. 


Fig.  62. 


56 


POWERS   AND   ROOTS 


12.  The  diagonal  of  a  square  is  8  inches,  find  the  side. 

Ans.  5.657  in. 

13.  Find    the  area  of  a  square  the    diagonal  of  which 
is  25  inches. 

Ans.  312.5  sq.  in. 

14.  Find  the  diagonal  of  the  square  nut  of  Fig.  63. 

Ans.  1.768  in. 


Fig.  64. 


15.  The  diagonal  of  a  square  nut,  Fig.  64  is  3.9  inches, 

find  the  side. 

Ans.  2.756  in. 


16. 


Find    the    diagonal    of    the 
hexagonal  nut  shown  in  Fig- 

65. 

Ans.  4.04  in. 


REVIEW  PROBLEMS 


57 


17.  A  rod  is  1  inch  in  diameter,  Fig.  66,  what  is  the  size 
o.''  the  largest  square  bar  that  can  be  cut  from  it? 

Ans.  .7071  in. 


Fig.  66. 


18.  Find  the  number  of  pounds  of  metal  removed  in 
cutting  the  largest  possible  square  bar  from  a  steel  rod^ 
3  inches  in  diameter,  and  30  inches  in  length. 

Ans.  21.8  pounds. 


EXERCISE  5.     REVIEW  PROBLEMS 


1. 


24  T. 


Fig.  67. 


In  Fig.  67  is  shown 
the  meshing  of  gears  of 
a  lathe.  If  Gear  I  makes 
100  revolutions  a  minute, 
how  many  revolutions 
does  Gear  VI  make  in  a 
minute?  The  power  is 
transmitted  from  Gear  I 
to  Gear  III  to  Gear  IV, 
etc.  Gear  II  is  used  for 
reverse  only. 

(Gears  II  and  III  do 
not  mesh  with  IV  at  same 
time.) 

■  Ans.  30. 


58 


POWERS   AND   ROOTS 


2.  In  the  above  problem  the  stock  turns  with  Gear  I, 
and  Gear  VI  turns  the  lead  screw,  the  feed  is  uV  inch.  How 
long  will  it  take  to  cut  18  inches  on  the  stock? 

Ans.  19i  minutes. 


3. 


In  Fig.  68  k  shown 
the  position  of  gears 
m  reverse.  Find  the 
gear  ratio. 

Ans.  5.1  to  1. 


Fig.  68. — Reverse;  gears  turn  in 
direction  of  arrows. 

4.  No.  3  Morse  taper  is  .602  inch  per  foot.  The  small 
diameter  of  a  No.  3  Morse  taper  is  .788  inch.  The  length 
is  3f  inches.     What  is  the  cUameter  of  the  large  end? 

Ans.  .970  in, 

5.  A  piece  of  material  8  inches  in  diameter  is  making 
2  revolutions  per  second.     Find  the  cutting  speed  in  feet 

per  minute. 

Ans.  251  ft. 

6.  In  electroplating  one  ampere  of  current  deposits 
.001118  gram  of  silver  per  second.  How  many  grams  will 
be  deposited  by  17.5  amperes  in  one  hour? 

Ans.  70.4. 

7.  1250  pounds  of  nickel  steel  for  gears  contain  the 
following  elements:  Iron,  119-4  pounds;  nickel,  43  75 
pounds;    manganese,  8.125  pounds;    carbon,  3.125  pounds; 


REVIEW  PROBLEMS 


59 


phosphorus,  .5   pound;  and  sulphur,   .5  pound.     Find  the 
percent  of  each  element  in  the  alloy. 

Ans.  95.52%;  3.5%;  .65%;  .25%;  .04%;  .04%. 

8.  Mr.  Smith  bought  a  house  of  Charles  Brown  for 
$8100  and  paid  $2500  in  cash.  The  contract  requires  the 
payment  of  $75  on  the  first  day  of  each  month.  When  the 
payment  is  made  March  1st,  interest  on  $5500  for  one 
month  at  the  rate  of  6%  per  year,  $27.50,  is  credited  in  the 
interest  column  of  the  contract.  The  remainder  of  the 
payment,  $47.50,  is  subtracted  from  the  $5500;  the  balance, 
$5452.50,  draws  interest  for  the  next  month. 

Credit  for  each  month's  payment  is  given  on  a  page 
of  the  contract  as  shown  below.  Complete  the  record 
up  to  August  1. 


Date 

Principal 
Payments 

Balance 

of 

Principal 

Interest 
Payments 
Rate  6% 

Paying 

Interest 

to 

Signature 

Payment 

on 
Delivery. 

$2500.00 

$5500.00 

March  1 

$47.50 

$5452,50 

$27.50 

April  1 

Charles  Brown 

April      1 

Mav      1 

1 

1 
June       1   j 

July       1 

August  1 

9.  Find  the  volume  of  the  casting  shown  in  Fig.  69. 

Ans.  10.78  cu.  in. 


60 


POWERS   AND   ROOTS 


Fig.  69. 


10.  Find  the  volume  of  the  piece  shown  in  Fig.  70. 


ROOTS  OF   NUMBERS 


61 


Table  I 


Frac- 
tion 

Dec. 
equiv. 

Square 

Square 
root 

Cube 

Cube 
root 

Circum. 
circle 

Area 
circle 

Ti 

.01563 

.000244 

.1250 

.000003815 

.2500 

.04909 

.000192 

T2 

.03125 

.000976 

.  1768 

.00003052 

.3150 

.09818 

.000767 

16 

.0625 

.003906 

.2500 

.0002442 

.3968 

.  1963 

.003068 

3 
T2 

.09375 

.008789 

.3062 

.0008240 

.4543 

.2945 

.006903 

i 

. 12500 

.015625 

.  3535 

.001953 

.5000 

.3927 

.01228 

A 

. 15625 

.02441 

.3953 

.003815 

.5386 

.4909 

.01916 

A 

. 18750 

.03516 

.4330 

.006592 

.5724 

.5890 

.02761 

3T 

.21875 

.04786 

.4677 

01047 

.6025 

.6872 

.03758 

1 

4 

.25000 

.06250 

.5000 

.01562 

.6300 

.7854 

.04909 

■5% 

.28125 

.07910 

.5303 

02225 

.6552 

.8836 

.06213 

1  6 

.3125 

.09766 

.5590 

.03052 

.6786 

.9817 

.07670 

Ai 
3  2 

.34375 

.11829 

.5863 

04062 

.7005 

1.080 

.09281 

.37500 

. 14063 

.6124 

05273 

.7211 

1 .  178 

.1104 

6 

.40625 

. 16504 

.6374 

.06705 

.7406 

1.276 

.  1296 

A 

.43750 

.19141 

.6614 

08374 

.7592 

1.374 

.1503 

H 

.46875 

.2197 

.6847 

1030 

.7768 

1.473 

.1726 

1 
2 

.50000 

.2500 

.7071 

.1250 

.7937 

1.571 

.1963 

M 

.53125 

.2822 

.7289 

.  1499 

.8099 

1.669 

.2217 

9 
T6 

.5625 

.3164 

.7500 

.  1780 

8255 

1.767 

.2485 

if 

.59375 

.3525 

.7706 

2093 

.8405 

1.865 

.2769 

5 
8 

.6250 

.3906 

.7906 

2441 

.8550 

1.963 

.3068 

fi 

.65625 

.4307 

.8101 

2826 

.8690 

2.062 

.3382 

.6875 

.4727 

.8292 

3250 

.8826 

2.160 

.3712 

II 

.71875 

.5166 

.8478 

3713 

.8958 

2.258 

.4057 

3 

4 

.7500 

.5625 

.8660 

4219 

.9086 

2.356 

.4418 

# 

.78125 

.6104 

.8839 

.4768 

.9210 

2.454 

.4794 

i3 
1  6 

.8125 

.6602 

.9014 

5364 

.9331 

2.553 

.5185 

2  7 

.84375 

.7119 

.9186 

6007 

.9449 

2.651 

.5592 

V 

8 

.8750 

.7656 

.9354 

.6699 

.9565 

2.749 

.6013 

2.9 

.90625 

.8213 

9520 

.7443 

.9677 

2.847 

.6450 

il 

.9375 

.8789 

.9682 

8240 

.9787 

2.945 

.6903 

p 

.96875 

.9385 

^43 

9091 

.9895 

3,043 

.7371 

f 

1 

1 

1 

1 

1 

3.1416 

.7854 

(yz 

ROOTS  OF   NU.MBERS 

Table  II 

No. 

Square 

Cube 

Square 
root 

Cube 
root 

Diam- 
eter 

Area 

Circum- 
ference 

1 

1 

1 

1.0000 

1 

1.0000 

1 

0,785 

3.142 

2 

4 

8 

1.4142 

1.2599 

2 

3.142 

6.283 

3 

9 

27 

1.7321 

1 . 4422 

3 

7.069 

9.425 

4 

16 

64 

2.0000 

1.5874 

4 

12.566 

12.566 

6 

25 

125 

2.2361 

1.7100 

5 

19.635 

15.708 

6 

36 

216 

2.4495 

1.8171 

6 

28.274 

18.850 

7 

49 

343 

2.6458 

1.9129 

7 

38.485 

21.991 

8 

64 

512 

2.8284 

2.0000 

8 

50.265 

25.133 

9 

81 

729 

3.0000 

2.0801 

9 

63.617 

28.274 

10 

100 

1,000 

3.1623 

2.1544 

10 

78.540 

31.416 

11 

121 

1,331 

3.3166 

2.2240 

11 

95.033 

34.558 

12 

144 

1,728 

3.4641 

2.2894 

12 

113.10 

37.699 

13 

169 

2,197 

3.6050 

2.3513 

13 

132  73 

40.841 

14 

196 

2,744 

3.7417 

2.4101 

14 

153 . 94 

43.982 

15 

225 

3,375 

3.8730 

2.4662 

15 

176.71 

47.124 

16 

256 

4,096 

4.0000 

2.5198 

16 

201 . 06 

50.266 

17 

289 

4,913 

4.1231 

2.5713 

17 

•  226.98 

53.407 

18 

324 

5,832 

4.2420 

2.6207 

18 

254.47 

56.549 

19 

361 

6,859 

4.3589 

2.6084 

19 

283.53 

59.690 

20 

400 

8,000 

4.4721 

2.7144 

20 

314.16 

62.832 

21 

441 

9,261 

4.5820 

2.7589 

21 

346.36 

65.973 

22 

484 

10,648 

4.6904 

2.8020 

22 

380.13 

69.115 

23 

529 

12,167 

4.795S 

2.8439 

23 

415.48 

72.257 

24 

576 

13.824 

4 .  8i)90 

2.8S45 

24 

452.39 

75.398 

25 

625 

15,625 

5.0000 

2.9240 

25 

490.87 

78.540 

20 

676 

17,576 

5.0990 

2.9025 

26 

530.93 

81.681 

27 

729 

19,683 

5.1902 

3.0000 

27 

572 . 56 

84.823 

28 

784 

21,952 

5.2915 

3.0366 

28 

615.75 

87.965 

29 

841 

24,389 

5.3852 

3.0723 

29 

660.52 

91 . 106 

30 

900 

27,000 

5.4772 

3.1072 

30 

706.86 

94.248 

31 

901 

29,791 

5.5078 

3.1414 

31 

754 . 77 

97 . 389 

32 

1024 

32,768 

5.6569 

3.1748 

32 

804 . 25 

100  53 

33 

1089 

35,9.37 

5.7446 

3.2075 

33 

855 . 30 

103.67 

34 

1156 

39,304 

5.8310 

3 . 2396 

34 

907.92 

106.81 

35 

1225 

42,875 

5.9161 

3.2711 

35 

962 . 1 1 

109.96 

36 

1296 

46,656 

6.0000 

3.3019 

36 

1017.9 

113.10 

37 

1369 

50,653 

6.0828 

3.3322 

37 

1075 . 2 

116.24 

38 

1444 

54,872 

6.1644 

3.3620 

38 

1131   1 

119.38 

39 

1521 

59,319 

6.2450 

3.3912 

39 

1191.6 

122.52 

40 

1600 

64,000 

6.3246 

3.4200 

40 

1256.6 

125.66 

ROOTS  OF  NUMBERS 

Table  II  —  Continued 


63 


No. 

Square 

Cube 

Square 
root 

Cube 
root 

Diam- 
eter 

Area 

Circum- 
ference 

41 

1681 

68,921 

6.4031 

3.4482 

41 

1320.3 

128.80 

42 

1764 

74,088 

6.4807 

3.4760 

42 

1385.4 

131.95 

43 

1849 

79,507 

6.5574 

3.5034 

43 

1452.2 

135.09 

44 

1936 

85,184 

6.6332 

3.5303 

44 

1520.5 

138.23 

45 

2025 

91,125 

6.7082 

3.5569 

45 

1590.4 

141.37 

46 

2116 

97,336 

6.7823 

3.5830 

46 

1661.9 

144  51 

47 

2209 

103,823 

6.8557 

3.6088 

47 

1734.9 

147.65 

48 

2304 

110,592 

6.9282 

3.6342 

48 

1809.6 

150.80 

49 

2401 

117,649 

7.0000 

3.6593 

49 

1885.7 

153.94 

50 

2500 

125,000 

7.0711 

3.6840 

50 

1963.5 

157.08 

51 

2601 

132,651 

7.1414 

3.7084 

51 

2042.8 

160.22 

62 

2704 

140,608 

7.2111 

3.7325 

52 

2123.7 

163.36 

53 

2809 

148,877 

7.2801 

3.7563 

53 

2206.2 

166.50 

54 

2916 

157,464 

7.3485 

3.7798 

54 

2290.2 

169.65 

55 

3025 

166,375 

7.4162 

3.8030 

55 

2375 . 8 

172 . 79 

56 

3136 

175,616 

7. 4833 

3.8259 

56 

2463.0 

175.93 

57 

3249 

185,193 

7.5498 

3.8485 

57 

2551 . 8 

179.07 

58 

3364 

195,112 

7.6158 

3.8709 

58 

2642.1 

182.21 

59 

3481 

205,379 

7.6811 

3.8930 

59 

2734.0 

185.35 

60 

3600 

216,000 

7.7460 

3.9149 

60 

2827.4 

188.50 

61 

3721 

226,981 

7.8102 

3.9365 

61 

2922.5 

191.64 

62 

3844 

238,328 

7.8740 

3.9579 

62 

3019.1 

194.78 

63 

3969 

250,047 

7.9373 

3.9791 

63 

3117.2 

197.92 

64 

4096 

262,144 

8.0000 

4.0000 

64 

3217.0 

201.16 

65 

4225 

274,625 

8.0623 

4.0207 

65 

3318.3 

204.20 

66 

4356 

287,496 

8.1240 

4.0412 

66 

3421.2 

207.34 

67 

4489 

300,763 

8.18.54 

4.0615 

67 

3525.7 

210.49 

68 

4624 

314,432 

8.2462 

4.0817 

68 

3631.7 

213.63 

69 

4761 

328,509 

8.3066 

4.1016 

69 

3739.3 

216.77 

70 

4900 

343,000 

8.3666 

4.1213 

70 

3848.5 

219.91 

71 

5041 

357.911 

8.4261 

4.1408 

71 

39.59.2 

223.05 

,  72 

5184 

373,248 

8.4853 

4.1602 

72 

4071.5 

226 . 20 

73 

5329 

389,017 

8.5440 

4.1793 

73 

4185.4 

229.34 

74 

5476 

405,224 

8.6023 

4.1983 

74 

4300.8 

232 . 48 

75 

5625 

421,875 

8.6603 

4.2172 

75 

4417.9 

235.62 

76 

5776 

438,976 

8.7178 

4.2358 

76 

4536.5 

238.76 

77 

5929 

456,533 

8.7750 

4.2543 

77 

4656.6 

241.90 

78 

6084 

474,552 

s.asis 

4.2727 

78 

4778.4 

245.04 

79 

6241 

493,039 

8.8882 

4.2908 

79 

4901 . 7 

248.19 

80 

6400 

512,000 

8.9443 

4.3089 

80 

5026.5 

251.32 

64 


ROOTS  OF   NUMBERS 

Table  II  —  Continued 


Square 


6,561 
6,724 

6,889 
7,056 
7,225 
7,396 
7,569 
7,744 
7,921 
8,100 

8,281 
8,464 
8,649 
8,836 
9,025 
9,216 
9,409 
0,604 
9,801 
10,000 

10,201 
10,404 
10,609 
10,816 
11,025 
11,236 
11,449 
11,664 
11,881 
12,100 

12,321 
12,544 
12,769 
12,996 
13,225 
13,456 
13,(),S9 
13,924 
14,161 
14,400 


Cube 


531,441 
551,368 
571,787 
592,704 
614,125 
636,056 
658,503 
681,472 
704,969 
729,000 

753,571 

778,688 
804,357 
830,584 
857,375 
884,736 
912,673 
941,192 
970,299 
1,000,000 

1,030,301 
1,061,208 
1,092,727 
1,124,864 
1,157,625 
1,191,016 
1,225,043 
1,259,712 
1,295,029 
1,331,000 

1,367,631 
1,404,928 
1,442,897 
1,481,544 
1,520,875 
1,560,896 
1,601,613 
1,643,032 
1,685,159 
1,728,000 


Square 
root 


9.0000 
9.0554 
9.1104 
9.1652 
9.2195 
9.2736 
9.3276 
9.3808 
9.4340 
9.4868 

9.5394 
9.5917 
9.6437 
9.6954 
9.7468 
9.7980 
9.8489 
9.8995 
9.9499 
10.0000 

10.0499 
10.0995 
10.1489 
10.1980 
10.2470 
10.2956 
10.3441 
10.3923 
10,4403 
10.4881 

10.5357 
10.5830 
10,6301 
10.6771 
10,7238 
10,7703 
10,8167 
10.8628 
10.9087 
10.9545 


Cube 
root 

Diam- 
eter 

Area 

4,3267 

81 

5,153 

4.3445 

82 

5,281 

4.3621 

83 

5,411 

4.3795 

84 

5,542 

4.3968 

85 

5,675 

4,4140 

86 

5,809 

4,4310 

87 

5,945 

4.4480 

88 

6,082 

4.4647 

89 

6,221 

4.4814 

90 

6,362 

4.4979 

91 

6,504 

4.5144 

92 

6,648 

4.5307 

93 

6,793 

4.5468 

94 

6,940 

4.5629 

95 

7,088 

4.5789 

96 

7,238 

4.5947 

97 

7,390 

4.6104 

98 

7,543 

4,6261 

99 

7,698 

4.6416 

100 

7,854 

4.6570 

101 

8,012 

4.6723 

102 

8,171 

4.6875 

103 

8,332 

4.7027 

104 

8,495 

4.7177 

105 

8,659 

4.7326 

10(5 

8,825 

4,7475 

107 

8,992 

4,7()22 

108 

9,161 

4,7769 

109 

9,331 

4.7914 

110 

9,503 

4.8059 

111 

9,677 

4.8203 

112 

9,852 

4,8.34() 

113 

10,029 

4,S4S8 

114 

10,207 

4 . 8()29 

115 

10,387 

4,8770 

116 

10,568 

4.8910 

117 

10,751 

4.9049 

118 

10,936 

4.9187 

119 

11,122 

4.9324 

120 

11,310 

Circum- 
ference 


254.5 
257.6 
260.8 
263,9 
267.0 
270.2 
273.3 
276.5 
279.6 
282.7 

285.9 
289,0 
292.2 
295,3 
298,5 
301,6 
304 , 7 
307.9 
311,0 
314.2 

317,3 
320,4 
323.6 
326.7 
329.9 
333.0 
336.2 
339.3 
342.4 
345.6 

348.7 
351.9 
355.0 
358.1 
361.3 
364,4 
367,6 
370.7 
373.9 
377.0 


ROOTS    OF    NUMBERS 

Table  II  —  Continued 


65 


Square 


14,641 
14,884 
15,129 
15,376 
15,625 
15,876 
16,129 
16,384 
16,641 
16,900 

17,161 
17,424 
17,689 
17,956 
18,225 
18,496 
18,769 
19,044 
19,321 
19,600 

19,881 
20,164 
20,449 
20,736 
21,025 
21,316 
21,609 
21,904 
22,201 
22,500 

22,801 
23,104 
23,409 
23,716 
24,025 
24,336 
24,649 
24,964 
25,281 
25,600 


Cube 


1,771,561 
1,815,848 
1,860,867 
1,906,624 
1,953,125 
2,000,376 
2,048,383 
2,097,152 
2,146,689 
2,197,000 

2,248,091 
2,299.968 
2,352,637 
2,406,104 
2,460,375 
2,515.456 
2,571,353 
2,628,072 
2,685,619 
2,744,000 

2,803,221 
2,863,288 
2,924,207 
2,985,984 
3,048,625 
3,112,136 
3,176,523 
3,241,792 
3,307,949 
3,375,000 

3,442,951 
3,511,808 
3,581,577 
3,652,264 
3,723,875 
3,796,416 
3,869,893 
3,944,312 
4,019,679 
4,096,000 


Square 
root 


11.0000 
11.0454 
11.0905 
11.1355 
11.1803 
11.2250 
11.2694 
11.3137 
11.3578 
11.4018 

11.4455 
11.4891 
11.5326 
11.5758 
11.6190 
11.6619 
11.7047 
11.7473 
11.7898 
11.8322 

11.8743 
11.9164 
11.9583 
12.0000 
12.0416 
12.0830 
12.1244 
12.1655 
12.2066 
12.2474 

12.2882 
12.3288 
12.3693 
12.4097 
12.4499 
12.4900 
12.5300 
12.5698 
12.6095 
12.6491 


Cube 
root 

Diam- 
eter 

Area 

4.9461 

121 

11,499 

4.9597 

122 

11,690 

4.9732 

123 

11,882 

4.9866 

124 

12,076 

5.0000 

125 

12,272 

5.0133 

126 

12,469 

5.0265 

127 

12,668 

5.0397 

128 

12,868 

5.0528 

129 

13,070 

5.0658 

130 

13,273 

5.0788 

131 

13,478 

5.0916 

132 

13,685 

5.1045 

133 

13,893 

5.1172 

134 

14,103 

5.1299 

135 

14,314 

5.1426 

136 

14,527 

5.1551 

137 

14,741 

5.1676 

138 

14,957 

5.1801 

139 

15,175 

5.1925 

140 

15,394 

5.2048 

141 

15,615 

5.2171 

142 

15,837 

5.2293 

143 

16,061 

5.2415 

144 

16,286 

5.2536 

145 

16,513 

5.2656 

146 

16,741 

5.2776 

147 

16,972 

5.2896 

148 

17,203 

5.3015 

149 

17,437 

5.3133 

150 

17,671 

5.3251 

151 

17,908 

5.3368 

152 

18,146 

5.3485 

153 

18,385 

5.3601 

154 

18,626 

5.3717 

155 

18,869 

5.3832 

156 

19,113 

5.3947 

157 

19,359 

5.4061 

158 

19,607 

5.4175 

159 

19,856 

5.4288 

160 

20,106 

Circum- 
ference 


380.1 
383.3 
386.4 
389.6 
392.7 
395.8 
399.0 
402.1 
405.3 
408.4 

411.5 

414.7 

417.8 

421.0 

424. 

427. 

430. 

433. 

436. 


1 
3 

.4 
.5 

.7 
439.8 


443.0 
446.1 
449.2 
452.4 
455.5 
458.7 
461.8 
465.0 
468.1 
471.2 

474.4 
477.5 
480.7 
483.8 
486.9 
490.1 
493.2 
496.4 
499.5 
502.7 


66 


ROOTS    OF    NUxMBERS 
Table  II  —  Concluded 


No. 

Square 

Cube 

Square 

Cube 

Diam- 

Circum- 

root 

root 

eter 

ference    . 

161 

25,921 

4,173,281 

12.6886 

5.4401 

161 

20,358 

505.8' 

162 

26,244 

4,251,528 

12.7279 

5.4514 

162 

20,612 

508.9 

163 

26,569 

4,330,747 

12.7671 

5.4626 

163 

20,867 

512.1 

164 

26,896 

4,410,944 

12.8062 

5.4737 

164 

21,124 

515.2 

165 

27,225 

4,492,125 

12.8452 

5.4848 

165 

21,382 

518.4 

166 

27,556 

4,574,296 

12.8841 

5.4959 

166 

21,642 

521.5 

167 

27,889 

4,657,463 

12.9228 

5.5069 

167 

21.904 

524.6 

168 

28,224 

4,741,632 

12.' 9615 

5.5178 

168 

22,167 

527,8 

169 

28,561 

4,826,809 

13,0000 

5.5288 

169 

22,432 

530.9 

170 

28,900 

4,913,000 

13.0384 

5.5397 

170 

22^698 

534.1 

171 

29.241 

5,000,211 

13.0767 

5.5505 

171 

22,966 

537.2 

172 

29,584 

5.088,448 

13.1149 

5.5613 

172 

23,235 

.540.3 

173 

29,929 

5,177,717 

13.1529 

5.5721 

173 

23,506 

543,5 

174 

30,276 

5,268,024 

13.1909 

5.5828 

174 

23,779 

546,6 

175 

30,625 

5,359,375 

13.2288 

5.5934 

175 

24,053 

.549.8 

176 

30,976 

5,451,776 

13.2665 

5,6041 

176 

24,328 

552.9 

177 

31,329 

5,545.233 

13.3041 

5.6147 

177 

24,600 

5.56.1 

178 

31,684 

5,639,752 

13.3417 

5.62.52 

178 

24,885 

559,2 

179 

32,041 

5,735,339 

13.3791 

5.6357 

179 

25,165 

562.3 

180 

32,400 

5,832,000 

13.4164 

5.6462 

180 

25,447 

565.5 

181 

32,761 

5,029,741 

13.4536 

5 . 6567 

181 

25,730 

568.6 

182 

33,124 

6.02s.,-,(is 

13.4907 

5,6671 

182 

26,010 

571.8 

183 

33,489 

6,12S,4S7 

13.5277 

5 . 6774 

183 

26,302 

574.9 

184 

33,856 

6,229,504 

13.5647 

5.6877 

184 

26,590 

578.0 

185 

34,225 

6,331,625 

13.6015 

5.6980 

185 

20.880 

581.2 

186 

34,596 

6,434,856 

13.6382 

5,7083 

186 

27,172 

584.3 

187 

34,969 

6,539,203 

13.6748 

5,7185 

187 

27,405 

587.5 

188 

35,344 

6,644,672 

13.7113 

5.7287 

188 

27,759 

500.6 

189 

35,721 

6,751,269 

13.7477 

5.7388 

189 

28,055 

593.8 

190 

36,100 

6,859,000 

13.7840 

5.7489 

190 

28,353 

596.9 

191 

36,481 

6,967,871 

13.8203 

5.7590 

101 

28,652 

000.0 

192 

36,864 

7,077,888 

13.8564 

5.7690 

102 

28,953 

003 . 2 

193 

37,249 

7,189,057 

13.8924 

5.7790 

193 

29,2.55 

000 , 3 

194 

37,636 

7,301,384 

13.9284 

5.7890 

194 

29,559 

009,5 

195 

38,025 

7,414,875 

13.9642 

5.7989 

195 

29,805 

012,6 

196 

38,416 

7,529,536 

14.0000 

5.8088 

196 

30,172 

015,7 

197 

38,809 

7,645,373 

14,03.57 

5.8186 

197 

30,481 

018.9 

198 

39,204 

7,762,392 

14.0712 

5.8285 

198 

30,791 

022.0 

199 

39,601 

7,880,599 

14.1067 

5.8383 

199 

31,103 

625,2 

200 

40.000 

8,000,000 

14.1421 

5.8480 

200 

31,416 

628.3 

INDEX 


Altitude,  16 
Area  of  circle,  45 

rectangle,  16 

square,  17 

triangle,  19 

Base,  16 

Circle,  44 

—  area,  45 

—  circumference,  44 

—  diameter,  44 

—  radius,  44 
Cutting  speed,  44 
Cylinder,  volume  of,  46 

Denominator,  2 
Diagonal,  54 

Efficiency,  39 
Exponent,  48 

Factor,  2 

Feed,  27 

Fractions,  common,  2 

— ,  — ,  addition,  4 

— ,  — ,  changed  to  decimal,  35 

— ,  — ,  definition,  2 

— ,  — ,  division,  20 

— ,  — ,  equivalent,  3 


Fractions,  common,  improper,  6 
— ,  — ,  multiplication,  15 

— ,  subtraction,  10 

— ,  terms,  2 

decimal,  30 

— ,  addition,  31 

— ,  definition,  30 

— ,  division,  33 

— ,  multiplication,  32 

— ,  subtraction,  31 

Gears,  28 

— ,  automobile,  29,  58 

Hypotenuse,  53 

Lead  of  threads,  25 

Least  coimnon  denominator,  5,  7 

Micrometer,  40 
Mixed  number,  6 

Numerator,  2 

Percentage,  37 
Perimeter,  definition,  16 
Pitch  of  threads,  25 
Powers  of  numbers,  48' 

Radical  sign,  49 
Rectangle,  area,  16 


67 


68 


INDEX 


Rectangle,  perimeter,  16 
Rectangular  solid,  volume,  46 

Screw  threads,  25 
Square,  area,  17 


Taper,  43 

Terms  of  a  fraction,  2 
Trial  divisor,  50 
Triangle,  area,  19 
— ,  right,  53 


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